LTS Estimation

If the data are contaminated in the X space, M estimation might yield improper results. It is better to use the high breakdown
value method. This example shows how you can use LTS estimation to deal with X-space contaminated data. The following data
set, hbk, is an artificial data set that was generated by Hawkins, Bradu, and Kass (1984).

Both ordinary least squares (OLS) estimation and M estimation (not shown here) suggest that observations 11 to 14 are outliers.
However, these four observations were generated from the underlying model, whereas observations 1 to 10 were contaminated.
The reason that OLS estimation and M estimation do not pick up the contaminated observations is that they cannot distinguish
good leverage points (observations 11 to 14) from bad leverage points (observations 1 to 10). In such cases, the LTS method
identifies the true outliers.

The following statements invoke the ROBUSTREG procedure and use the LTS estimation method:

Figure 84.12 displays the model-fitting information and summary statistics for the response variable and independent covariates.

Figure 84.12: Model-Fitting Information and Summary Statistics

The ROBUSTREG Procedure

Model Information

Data Set

WORK.HBK

Dependent Variable

y

Number of Independent Variables

3

Number of Observations

75

Method

LTS Estimation

Summary Statistics

Variable

Q1

Median

Q3

Mean

StandardDeviation

MAD

x1

0.8000

1.8000

3.1000

3.2067

3.6526

1.9274

x2

1.0000

2.2000

3.3000

5.5973

8.2391

1.6309

x3

0.9000

2.1000

3.0000

7.2307

11.7403

1.7791

y

-0.5000

0.1000

0.7000

1.2787

3.4928

0.8896

Figure 84.13 displays information about the LTS fit, which includes the breakdown value of the LTS estimate. The breakdown value is a
measure of the proportion of contamination that an estimation method can withstand and still maintain its robustness. In this
example the LTS estimate minimizes the sum of 57 smallest squares of residuals. It can still estimate the true underlying
model if the remaining 18 observations are contaminated. This corresponds to a breakdown value around 0.25, which is set as
the default.

Figure 84.13: LTS Profile

LTS Profile

Total Number of Observations

75

Number of Squares Minimized

57

Number of Coefficients

4

Highest Possible Breakdown Value

0.2533

Figure 84.14 displays parameter estimates for covariates and scale. Two robust estimates of the scale parameter are displayed. For information
about computing these estimates, see the section Final Weighted Scale Estimator. The weighted scale estimator (Wscale) is a more efficient estimator of the scale parameter.

Figure 84.14: LTS Parameter Estimates

LTS Parameter Estimates

Parameter

DF

Estimate

Intercept

1

-0.3431

x1

1

0.0901

x2

1

0.0703

x3

1

-0.0731

Scale (sLTS)

0

0.7451

Scale (Wscale)

0

0.5749

Figure 84.15 displays outlier and leverage-point diagnostics. The ID variable index is used to identify the observations. If you do not specify this ID variable, the observation number is used to identify
the observations. However, the observation number depends on how the data are read. The first 10 observations are identified
as outliers, and observations 11 to 14 are identified as good leverage points.

Figure 84.15: Diagnostics

Diagnostics

Obs

index

Mahalanobis Distance

Robust MCD Distance

Leverage

StandardizedRobust Residual

Outlier

1

1

1.9168

29.4424

*

17.0868

*

2

2

1.8558

30.2054

*

17.8428

*

3

3

2.3137

31.8909

*

18.3063

*

4

4

2.2297

32.8621

*

16.9702

*

5

5

2.1001

32.2778

*

17.7498

*

6

6

2.1462

30.5892

*

17.5155

*

7

7

2.0105

30.6807

*

18.8801

*

8

8

1.9193

29.7994

*

18.2253

*

9

9

2.2212

31.9537

*

17.1843

*

10

10

2.3335

30.9429

*

17.8021

*

11

11

2.4465

36.6384

*

0.0406

12

12

3.1083

37.9552

*

-0.0874

13

13

2.6624

36.9175

*

1.0776

14

14

6.3816

41.0914

*

-0.7875

Figure 84.16 displays the final weighted least squares estimates. These estimates are least squares estimates that are computed after
the detected outliers are deleted.